Berry's geometric phase [1]} is an integral of a connection
along a quantum adiabatic cycle. Nontrivial values arise when the
curve corresponding to the adiabatic cycle encircles a degeneracy
point. The geometric phase can be viewed [2]} as the first
member of a series of cumulants extracted from the evolution of the
wave function along the adiabatic cycle. The higher order cumulants
are not geometric, they depend on the parametrization of the cycle,
not only on its geometry, but they are also physically well-defined.
We show that particular ratios of the cumulants give finite values for
cycles crossing degeneracy points. When the path crosses a degeneracy
point, the Berry phase becomes undefined, the higher order cumulants
all diverge. The machinery can not be applied in this case. In this
work we complement the formalism of the geometric phase by
constructing quantities which are meaningful in the case of paths
which cross degeneracy (or gap closure) points.
